We study the symmetries of nonrelativistic systems with an emphasis on applications to the fractional quantum Hall effect. A source for the energy current of a Galilean system is introduced and the nonrelativistic diffeomorphism invariance studied in previous work is enhanced to a full spacetime symmetry, allowing us to derive a number of Ward identities. These symmetries are smooth in the massless limit of the lowest Landau level. We develop a formalism for Newton-Cartan geometry with torsion to write these Ward identities in a covariant form. Previous results on the connection between Hall viscosity and Hall conductivity are reproduced.

There is significant recent work on coupling matter to Newton-Cartan spacetimes with the aim of investigating certain condensed matter phenomena. To this end, one needs to have a completely general spacetime consistent with local non-relativisitic symmetries which supports massive matter fields. In particular, one can not impose a priori restrictions on the geometric data if one wants to analyze matter response to a perturbed geometry. In this paper we construct such a Bargmann spacetime in complete generality without any prior restrictions on the fields specifying the geometry. The resulting spacetime structure includes the familiar Newton-Cartan structure with an additional gauge field which couples to mass. We illustrate the matter coupling with a few examples. The general spacetime we construct also includes as a special case the covariant description of Newtonian gravity, which has been thoroughly investigated in previous works. We also show how our Bargmann spacetimes arise from a suitable non-relativistic limit of Lorentzian spacetimes. In a companion paper [1] we use this Bargmann spacetime structure to investigate the details of matter couplings, including the Noether-Ward identities, and transport phenomena and thermodynamics of nonrelativistic fluids.Recently there has been a revival of interest in the Newton-Cartan description of nonrelativistic spacetimes in the condensed matter literature [2][3][4][5][6][7][8][9][10][11][12] where, it has been used with great effect to describe phenomena in the quantum Hall effect and various transport phenomena in condensed matter systems. Newton-Cartan spacetimes are used to describe matter fields and their interaction with general background geometries which are consistent with non-relativistic Galilean invariance. Newton-Cartan geometry also arises in the study of nonrelativistic holographic systems, where the boundary theory realizes a "twistless-torsionful" Newton-Cartan geometry [13][14][15][16][17][18][19]. On the other hand, in the gravitational physics literature (see [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] and also Ch.12 of [35] and Ch.4 of [36]) Newton-Cartan geometry has been well studied as a diffeomorphism-covariant, geometric way to describe Newtonian gravity. As such, the Newton-Cartan spacetimes considered there belong to a much more restricted class. This divergence of interests has lead to some conflicts in the construction (or at least in the interpretation) of Newton-Cartan spacetimes. One of the aims of this work is to alleviate these conflicts and set a clear stage for describing both Newtonian gravity and matter R I J := dω I J + ω I K ∧ ω K J (2.13b) 7Here we use the convention that the connection is a 1-form valued in the Lie algebra, instead of viewing it as 1-form components in a set of bases given by the generators of the Lie algebra.

We consider non-relativistic curved geometries and argue that the background structure should be generalized from that considered in previous works. In this approach the derivative operator is defined by a Galilean spin connection valued in the Lie algebra of the Galilean group. This includes the usual spin connection plus an additional "boost connection" which parameterizes the freedom in the derivative operator not fixed by torsion or metric compatibility. As an example we write down the most general theory of dissipative fluids consistent with the second law in curved non-relativistic geometries and find significant differences in the allowed transport coefficients from those found previously. Kubo formulas for all response coefficients are presented. Our approach also immediately generalizes to systems with independent mass and charge currents as would arise in multicomponent fluids. Along the way we also discuss how to write general locally Galilean invariant non-relativistic actions for multiple particle species at any order in derivatives. A detailed review of the geometry and its relation to non-relativistic limits may be found in a companion paper.

In this paper we investigate properties of Chern-Simons theory coupled to massive fermions in the large N limit. We demonstrate that at low temperatures the system is in a Fermi liquid state whose features can be systematically compared to the standard phenomenological theory of Landau Fermi liquids. This includes matching microscopically derived Landau parameters with thermodynamic predictions of Landau Fermi liquid theory. We also calculate the exact conductivity and viscosity tensors at zero temperature and finite chemical potential. In particular we point out that the Hall conductivity of an interacting system is not entirely accounted for by the Berry flux through the Fermi sphere. Furthermore, investigation of the thermodynamics in the non-relativistic limit reveals novel phenomena at strong coupling. As the 't Hooft coupling λ approaches 1, the system exhibits an extended intermediate temperature regime in which the thermodynamics is described by neither the quantum Fermi liquid theory nor the classical ideal gas law. Instead, it can be interpreted as a weakly coupled quantum Bose gas.